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The Average Tree Solution for Cooperative Games with Communication Structure

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Author Info

  • P. Jean-Jacques Herings

    ()
    (Maastricht University)

  • Gerard van der Laan

    ()
    (VU University Amsterdam)

  • Dolf Talman

    ()
    (Tilburg University)

  • Zaifu Yang

    ()
    (Yokohama National University)

Abstract

We study cooperative games with communication structure, represented by an undirected graph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class of games. Given the graph structure we define a collection of spanning trees, where each spanning tree specifies a particular way by which players communicate and determines a payoff vector of marginal contributions of all the players. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has a complete communication structure, then the proposed solution coincides with the Shapley value, and that if the game has a cycle-free communication structure, it is the solution proposed by Herings, van der Laan and Talman (2008). We introduce the notion of link-convexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 08-083/1.

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Date of creation: 04 Sep 2008
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Handle: RePEc:dgr:uvatin:20080083

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Web page: http://www.tinbergen.nl

Related research

Keywords: Cooperative game; graph structure; single-valued solution; core; convexity; spanning tree;

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References

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  1. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print halshs-00581662, HAL.
  2. Le Breton, M. & Owen, G. & Weber, S., 1991. "Strongly Balanced Cooperative Games," Papers 92-3, York (Canada) - Department of Economics.
  3. HERINGS, P. Jean-Jacques & van der LAAN, Gerard & TALMAN, Dolf, . "The average tree solution for cycle-free graph games," CORE Discussion Papers RP -2155, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  4. Talman, A.J.J. & Yamamoto, Y., 2007. "Games With Limited Communication Structure," Discussion Paper 2007-19, Tilburg University, Center for Economic Research.
  5. René van den Brink & Gerard van der Laan & Vitaly Pruzhansky, 2004. "Harsanyi Power Solutions for Graph-restricted Games," Tinbergen Institute Discussion Papers 04-095/1, Tinbergen Institute.
  6. Mamoru Kaneko & Myrna Holtz Wooders, 1982. "Cores of Partitioning Games," Cowles Foundation Discussion Papers 620, Cowles Foundation for Research in Economics, Yale University.
  7. Demange, G., 1991. "Intermediate Preferences and Stable Coalition Structures," DELTA Working Papers 91-16, DELTA (Ecole normale supérieure).
  8. Marco Slikker, 2005. "A characterization of the position value," International Journal of Game Theory, Springer, vol. 33(4), pages 505-514, November.
  9. René van den Brink & Gerard van der Laan & Vitaly Pruzhansky, 2004. "Harsanyi Power Solutions for Graph-restricted Games," Tinbergen Institute Discussion Papers 04-095/1, Tinbergen Institute.
  10. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.
  11. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154855, Tilburg University.
  12. Talman, A.J.J. & Yamamoto, Y., 2008. "Average tree solution and subcore for acyclic graph games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3129818, Tilburg University.
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Citations

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Cited by:
  1. Rene van den Brink & Gerard van der Laan & Nigel Moes, 2012. "A Strategic Implementation of the Average Tree Solution for Cycle-Free Graph Games," Tinbergen Institute Discussion Papers 12-050/1, Tinbergen Institute.
  2. Sylvain Béal & Eric Rémila & Philippe Solal, 2012. "Compensations in the Shapley value and the compensation solutions for graph games," International Journal of Game Theory, Springer, vol. 41(1), pages 157-178, February.
  3. Koshevoy, G.A. & Suzuki, T. & Talman, A.J.J., 2013. "Solutions For Games With General Coalitional Structure And Choice Sets," Discussion Paper 2013-012, Tilburg University, Center for Economic Research.
  4. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.
  5. Koshevoy, G.A. & Talman, A.J.J., 2011. "Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)," Discussion Paper 2011-119, Tilburg University, Center for Economic Research.
  6. Selçuk, Özer & Suzuki, Takamasa & Talman, Dolf, 2013. "Equivalence and axiomatization of solutions for cooperative games with circular communication structure," Economics Letters, Elsevier, vol. 121(3), pages 428-431.
  7. repec:pdn:wpaper:73 is not listed on IDEAS
  8. Suzuki, T. & Talman, A.J.J., 2011. "Solution Concepts for Cooperative Games with Circular Communication Structure," Discussion Paper 2011-100, Tilburg University, Center for Economic Research.
  9. Napel, Stefan & Nohn, Andreas & Alonso-Meijide, José Maria, 2012. "Monotonicity of power in weighted voting games with restricted communication," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 247-257.
  10. Koshevoy, Gleb & Talman, Dolf, 2014. "Solution concepts for games with general coalitional structure," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 19-30.
  11. S. Ryuo & K. Sato & Y. Yamamoto, 2012. "Parameterized fairness axioms on cycle-free graph games," Journal of Global Optimization, Springer, vol. 52(3), pages 487-497, March.
  12. Huseynov, T. & Talman, A.J.J., 2012. "The Communication Tree Value for TU-games with Graph Communication," Discussion Paper 2012-095, Tilburg University, Center for Economic Research.
  13. Khmelnitskaya, A. & Selcuk, O. & Talman, A.J.J., 2012. "The Average Covering Tree Value for Directed Graph Games," Discussion Paper 2012-037, Tilburg University, Center for Economic Research.
  14. Khmelnitskaya, Anna B. & Sudhölter, Peter, 2011. "The prenucleolus for games with communication structures," Discussion Papers of Business and Economics 10/2011, Department of Business and Economics, University of Southern Denmark.
  15. Rene van den Brink & Gerard van der Laan & Nigel Moes, 2012. "A Strategic Implementation of the Average Tree Solution for Cycle-Free Graph Games," Tinbergen Institute Discussion Papers 12-050/1, Tinbergen Institute.
  16. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2009. "Average tree solutions and the distribution of Harsanyi dividends," MPRA Paper 17909, University Library of Munich, Germany.
  17. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2009. "Weighted Component Fairness for Forest Games," MPRA Paper 17455, University Library of Munich, Germany.
  18. Albizuri, M. Josune & Sudhölter, Peter, 2014. "On the Core of Games with Communication Structures," Discussion Papers of Business and Economics 6/2014, Department of Business and Economics, University of Southern Denmark.

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